Exercise 3.2.9 (Primes $q$ such that $\genfrac{(}{)}{}{}{5}{q} = -1$)

Question

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

Find all primes $q$ such that $\legendre{5}{q} = -1$.

Richard Ganaye · Accepted Answer

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\begin{proof} Since $5 \equiv 1 \pmod 4$, $\legendre{5}{q} = \legendre{q}{5}$. Therefore $\legendre{5}{q} = -1$ if and only if $q$ is a quadratic nonresidue modulo $5$:

$$\legendre{5}{q} = -1 \iff q \equiv 2, 3 \pmod 5.$$
\end{proof}

Exercise 3.2.9 (Primes $q$ such that $\genfrac{(}{)}{}{}{5}{q} = -1$)

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Exercise 3.2.9 (Primes $q$ such that $\genfrac{(}{)}{}{}{5}{q} = -1$)

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